reserve a,b,c,d for Real;
reserve z,z1,z2 for Complex;

theorem
  Re z1 = 0 & Re z2 = 0 & Im z2 <> 0 implies Re(z1/z2) = (Im z1)/(Im z2)
  & Im(z1/z2) = 0
proof
  assume that
A1: Re z1 = 0 and
A2: Re z2 = 0 & Im z2 <> 0;
A3: z1/z2 = z1*z2" & Re(z2") = 0 by A2,Th23,XCMPLX_0:def 9;
  hence Re(z1/z2) = -(Im z1)*Im(z2") by A1,Th15
    .= -(Im z1)*-(Im z2)" by A2,Th23
    .= --(Im z1)*(Im z2)"
    .= (Im z1)/(Im z2) by XCMPLX_0:def 9;
  thus thesis by A1,A3,Th15;
end;
